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The printed root characterization of a wall-crossing element

PDF p. 10: paragraph after (3.15)

Statement

Let $(W, S)$ be a Coxeter system indexed by the vertex set $Γ_{0}$ . For $L \subseteq Γ_{0}$ , let $W_{L}$ be the subgroup generated by the simple reflections indexed by $L$ , let $Φ_{L}$ be its associated root set, and put $Φ_{L}^{\pm} = Φ_{L} \cap Φ^{\pm}$ . For a chamber label $(x, I) \in C h a m (J)$ and $a \in Γ_{0} ∖ I$ , suppose $W_{D}$ is finite, where $D = [I, a]$ is the connected component containing $a$ in the Coxeter diagram induced on $I \cup {a}$ . Let $w_{L}$ denote the longest element of a finite $W_{L}$ , let $ι_{D}$ be defined by $w_{D} α_{i} = - α_{ι_{D} (i)}$ , and set $v_{a, I} = w_{D ∖ {a}} w_{D}$ and $K = (I \cup {a}) ∖ {ι_{D} (a)}$ .

The article claims that $v_{a, I}$ is the unique element satisfying $v_{a, I} (Φ_{I \cup {a}}^{+} ∖ Φ_{K}^{+}) \subseteq Φ_{K}^{-}$ and $v_{a, I} (Φ_{K}^{+}) \subseteq Φ_{I \cup {a}}^{+}$ . This claim is false as printed: in the rank-one case below, $v_{a, I}$ fails the first inclusion.

Statement references

Coxeter-system, roots, length and weak-order conventions

Standard parabolic subgroups and parabolic roots

Algebraic chamber-label set

Connected component notation [I,a]

Longest parabolic elements and opposition permutation

Simple wall-crossing elements and maps

Proof

Given: The rank-one Coxeter system with $Γ_{0} = {a}$ and generator $s_{a}$ , and $J = I = \emptyset$ , $x = 1$ .

[F1] The standard parabolic subgroup $W_{L}$ is generated by the simple reflections indexed by $L$ ; $Φ_{L} = {w α_{i} : w \in W_{L}, i \in L}$ and $Φ_{L}^{\pm} = Φ_{L} \cap Φ^{\pm}$ . Standard parabolic subgroups and parabolic roots

[F2] The simple root $α_{a}$ is a nonzero basis vector in the positive cone, and $s_{a} α_{a} = - α_{a}$ . Also $s_{a}^{2} = 1$ . Coxeter-system, roots, length and weak-order conventions

[F3] A chamber label $(x, I)$ belongs to $C h a m (J)$ when $W_{J} x = x W_{I}$ and $x$ has minimum length in $x W_{I}$ . Algebraic chamber-label set

[F4] The set $[I, a]$ is the component containing $a$ of the diagram induced on $I \cup {a}$ . Connected component notation [I,a]

[F5] For finite $W_{L}$ , $w_{L}$ is its longest element; $ι_{L}$ is the permutation satisfying $w_{L} α_{i} = - α_{ι_{L} (i)}$ . In particular $w_{\emptyset} = 1$ . Longest parabolic elements and opposition permutation

[F6] The wall-crossing element is $v_{a, I} = w_{D ∖ {a}} w_{D}$ , and $K = (I \cup {a}) ∖ {ι_{D} (a)}$ . Simple wall-crossing elements and maps

Proof technique: direct.

Goal: verify the article's hypotheses in this example and show that its first inclusion fails.

1.1F1F2F3F4given

Since $W_{\emptyset} = {1}$ , the two cosets in the chamber-label condition are both ${1}$ , and $1$ has minimum length in its coset. Thus $(1, \emptyset) \in C h a m (\emptyset)$ . The induced diagram has the single vertex $a$ , so $D = {a}$ . The subgroup $W_{D} = {1, s_{a}}$ is finite; $s_{a}$ is nontrivial because it sends the nonzero vector $α_{a}$ to its negative. Hence all the stated hypotheses hold.

2.1F1F2F5F6step 1.1∎

The longest element of $W_{D}$ is $s_{a}$ : the identity has length zero and $s_{a}$ has length one. Consequently $ι_{D} (a) = a$ , $K = \emptyset$ , and $v_{a, I} = 1 \cdot s_{a} = s_{a}$ . By the root-set definition, $Φ_{K}^{+} = Φ_{K}^{-} = \emptyset$ , while $α_{a} \in Φ_{{a}}^{+}$ . Thus the left side of the first claimed inclusion contains $v_{a, I} α_{a} = - α_{a}$ , but its right side is empty. The claimed inclusion, and hence the asserted characterization, fails.

Source notes

The counterexample addresses the formula as printed. It does not establish a general replacement characterization.

Depends on

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 10, paragraph immediately after (3.15)